We report on gamma-ray observations in the off-pulse window of the Vela pulsar PSR B0833-45, using 11 months of survey data from the Fermi Large Area Telescope (LAT). This pulsar is located in the 8 degree diameter Vela supernova remnant, which contains several regions of non-thermal emission detected in the radio, X-ray and gamma-ray bands. The gamma-ray emission detected by the LAT lies within one of these regions, the 2*3 degrees area south of the pulsar known as Vela-X. The LAT flux is signicantly spatially extended with a best-fit radius of 0.88 +/- 0.12 degrees for an assumed radially symmetric uniform disk. The 200 MeV to 20 GeV LAT spectrum of this source is well described by a power-law with a spectral index of 2.41 +/- 0.09 +/- 0.15 and integral flux above 100 MeV of (4.73 +/- 0.63 +/- 1.32) * 10^{-7} cm^{-2} s^{-1}. The first errors represent the statistical error on the fit parameters, while the second ones are the systematic uncertainties. Detailed morphological and spectral analyses give strong constraints on the energetics and magnetic field of the pulsar wind nebula (PWN) system and favor a scenario with two distinct electron populations.
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